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\textbf{Transformation of Fokker-Planck Equation from
(\emph{E,}}\(\mathbf{\ }\mathbf{\alpha}_{\mathbf{0}}\)\textbf{) to
(\emph{V}, \emph{K})}

\textbf{1. Diffusion Equation in \emph{V} and \emph{K}}

\[\frac{\partial f}{\partial t} = \frac{1}{\hat{G}}\frac{\partial}{\partial V}\hat{G}\left\lbrack D_{VV}\frac{\partial f}{\partial V} + D_{VK}\frac{\partial f}{\partial K} \right\rbrack + \frac{1}{\hat{G}}\frac{\partial}{\partial K}\hat{G}\left\lbrack D_{KV}\frac{\partial f}{\partial V} + D_{KK}\frac{\partial f}{\partial K} \right\rbrack\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (1)\]

where

\[V = M\left( K + a \right)^{b}\]

\[\hat{G} = \left( K + a \right)^{- 3b/2}\sqrt{V}\]

\[D_{VV} = \left( K + a \right)^{2b}\operatorname{}\alpha_{0}\left\lbrack \left( \frac{E(E + E_{0})\sin\alpha_{0}}{E_{0}B_{0}} \right)^{2}\frac{D_{EE}}{E^{2}} + \left( \frac{E\left( E + 2E_{0} \right)\cos\alpha_{0}}{E_{0}B_{0}} \right)^{2}D_{\alpha_{0}\alpha_{0}} + 2\left( \frac{E + E_{0}}{E_{0}B_{0}} \right)\left( \frac{E^{2}\left( E + 2E_{0} \right)\sin\alpha_{0}\cos\alpha_{0}}{E_{0}B_{0}} \right)\frac{D_{\alpha_{0}E}}{E} \right\rbrack\]

\[D_{VK} = {- \left( K + a \right)}^{b}\ 2R_{0}T\left( \alpha_{0} \right)\left\lbrack \left( \frac{E\left( E + 2E_{0} \right)}{E_{0}\sqrt{B_{0}}} \right)\left( \frac{\operatorname{}\alpha_{0}}{\sin\alpha_{0}}\  \right)D_{\alpha_{0}\alpha_{0}} + \ \left( \frac{E(E + E_{0})}{E_{0}\sqrt{B_{0}}} \right)\cos\alpha_{0}\frac{D_{\alpha_{0}E}}{E} \right\rbrack\]

\[D_{KK} = B_{0}\left( \frac{\cos\alpha_{0}}{\operatorname{}\alpha_{0}}\ 2R_{0}T\left( \alpha_{0} \right) \right)^{2}D_{\alpha_{0}\alpha_{0}}\]

where

\emph{m} = rest mass

\[T\left( \alpha_{0} \right) = \frac{1}{2R_{0}}\int_{s_{m}}^{s_{m'}}\frac{ds}{\cos\mathrm{\alpha}}\]

\textbf{2. Diffusion Equation in ln\emph{V} and ln\emph{K}}

Let \emph{U\textsubscript{V}} = ln\emph{V}, \emph{U\textsubscript{K}} =
ln\emph{K} , then (1) becomes

\[\frac{\partial f}{\partial t} = \frac{1}{G_{U}}\frac{\partial}{\partial U_{V}}\left\lbrack G_{U}\left( \frac{D_{VV}}{V^{2}}\frac{\partial f}{\partial U_{V}} + \frac{D_{VK}}{VK}\frac{\partial f}{\partial U_{K}} \right) \right\rbrack + \frac{1}{G_{U}}\frac{\partial}{\partial U_{K}}\left\lbrack G_{U}\left( \frac{D_{\text{KV}}}{VK}\frac{\partial f}{\partial U_{V}} + \frac{D_{KK}}{K^{2}}\frac{\partial f}{\partial U_{K}} \right) \right\rbrack\text{\ \ \ }\text{\ \ \ \ \ \ \ \ \ \ }(2)\]

where

\[G_{U} = K{(K + a)}^{- 3b/2}V^{3/2}\]
